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1
Make salsa: Preheat oven to 400u00b0. Line a baking sheet with foil and top with tomatoes, onion, garlic, and jalapeno. Bake until softened, about 25 minutes, turning ingredients halfway through baking. Remove from oven and let cool.
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2
Core tomatoes, peel onion and garlic, and stem jalapeno. Put the vegetables, lime juice, cilantro, and salt in a food processor and pulse just until chunky and incorporated. Transfer salsa to a bowl.
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3
Prepare chiles: Preheat broiler. Arrange poblanos on foil-lined baking sheet in a single layer and broil 2 to 3 in. from heat until blackened, turning as needed, 10 to 12 minutes. Put chiles in a bowl, cover, and let sit 10 minutes to loosen skins.
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4
Peel chiles, pull out stems (keeping chiles as intact as possible), and remove seeds with a spoon or your fingers. Blot with paper towels. Preheat oven to 200u00b0.
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5
In a bowl, combine chopped onion and shredded cheese. Squeeze handfuls to form 6 narrow logs. Carefully stuff logs into chiles through stem ends.
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6
Pour oil into a large pot to a depth of 2 to 3 in. and heat to 375u00b0. Meanwhile, separate 5 eggs. In the bowl of a mixer, beat whites until soft peaks form. In another bowl, whisk yolks with 1/3 cup flour and 1/4 tsp. salt. Add a spoonful of whites to yolk mixture and whisk to combine, then gently fold yolk mixture into whites just until incorporated.
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7
Put 1/2 cup flour in a shallow bowl. Using tongs or your hands, dip a chile first into flour, then into egg batter, turning gently to coat. Gently lower chile into hot oil. Repeat with a second chile. Cook until browned, gently turning once with tongs, about 4 minutes total. Transfer chiles to a baking sheet lined with paper towels and keep warm in the oven. Cook remaining chiles the same way.
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8
To poach eggs, fill a wide, deep frying pan with enough water to cover an egg out of the shell by about 1 in. Heat water until 1 or 2 bubbles break the surface. Reduce heat until bubbles form on pan bottom and occasionally pop to the surface. Break open remaining 6 eggs, one at a time, into water, keeping eggs close to the surface. Cook until eggs are done the way you like, 3 to 5 minutes for soft yolks with firm whites or 7 to 10 minutes for firm yolks and whites. Remove eggs from water with a slotted spoon and transfer to a plate.
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9
Divide chiles among 6 plates, top each with a poached egg and a spoonful of salsa, and serve remaining salsa on the side.
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10
Make ahead: Salsa keeps 1 day, covered and chilled.
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11
Note: Nutritional analysis is per serving.
